Places worth visiting overnight beyond Budapest

We visited Bratislava between Budapest & Vienna. It's only 39 miles from Vienna, but it's a whole word apart. In a weekend there, we didn't hear a word of German. Just English and Hungarian. It's a lovely river town.

I enjoyed Pecs a lot. I also liked the smaller town of Gyor to the west and Kecskemet and Szeged to the south (and east of Pecs, but I don't think there's any direct transportation from Pecs). Gyor gets you within bus distance of the Pannonhalma Archabbey, and Gyor itself has a pretty historic center with a lot of small museums (many of them art-related). Pecs, Kecskemet and Szeged all have interesting architecture.

Pecs is the home of the Zsolnay ceramics company, which in the first half of the 20th century was prominent in the field or architectural ceramics that would withstand outdoor use. The company also produced decorative objects that are well represented in local museums.

From Szeged one can take a bus across the border to Subotica in Serbia. Those two cities have a lot of Art Nouveau architecture. Subotica's wasn't as well-kept as what I saw in Szeged (this was in 2015).

I will add that Budapest is a large, very appealing city. You may find you don't want to leave after 4 nights. Or 5 nights...

Eger, Pecs both worth at least one night. Early train down, late train back. Or two nights.....
Szentendre is a good day trip as is Visegrad or Esztergom (some try all three in one day ... a bit much really)
Vac is another good day trip.
And I've had a few enjoyable overnight stays in the local bars

I took the train from Budapest to Pecs for 2 nights, then the bus to Szeged for 1 night, then the train to Kecskemét for a night and back to Budapest by train. I left most of my luggage in Budapest at my hotel and just took a backpack. I enjoyed all 3 towns and felt that the time I spent in each was appropriate. I especially liked the Art Nouveau architecture. My trip was in late September when days were long and weather was warm.

And I have spotted Chani in at leat one bar.

We overnighted in Szeged on the way to Serbia (its practically on the boarder). Nice train ride of a few hours - we really liked the small laid-back town with excellent coffee shops and a nice museum which we didn't go to because we spent too much time in the coffee shop.